View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Universidade do Minho: RepositoriUM The inverse along a lower triangular matrix∗ Xavier Marya, Pedro Patr´ıciob aUniversit´eParis-Ouest Nanterre { La D´efense,Laboratoire Modal'X, 200 avenuue de la r´epublique,92000 Nanterre, France. email:
[email protected] bDepartamento de Matem´aticae Aplica¸c~oes,Universidade do Minho, 4710-057 Braga, Portugal. email:
[email protected] Abstract In this paper, we investigate the recently defined notion of inverse along an element in the context of matrices over a ring. Precisely, we study the inverse of a matrix along a lower triangular matrix, under some conditions. Keywords: Generalized inverse, inverse along an element, Dedekind-finite ring, Green's relations, rings AMS classification: 15A09, 16E50 1 Introduction In this paper, R is a ring with identity. We say a is (von Neumann) regular in R if a 2 aRa.A particular solution to axa = a is denoted by a−, and the set of all such solutions is denoted by af1g. Given a−; a= 2 af1g then x = a=aa− satisfies axa = a; xax = a simultaneously. Such a solution is called a reflexive inverse, and is denoted by a+. The set of all reflexive inverses of a is denoted by af1; 2g. Finally, a is group invertible if there is a# 2 af1; 2g that commutes with a, and a is Drazin invertible if ak is group invertible, for some non-negative integer k. This is equivalent to the existence of aD 2 R such that ak+1aD = ak; aDaaD = aD; aaD = aDa.